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Journal of Turbulence

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Dynamos at extreme magnetic Prandtl numbers:insights from shell models

Mahendra K. Verma & Rohit Kumar

To cite this article: Mahendra K. Verma & Rohit Kumar (2016) Dynamos at extreme magneticPrandtl numbers: insights from shell models, Journal of Turbulence, 17:12, 1112-1141, DOI:10.1080/14685248.2016.1228937

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JOURNAL OF TURBULENCE, VOL. , NO. , –http://dx.doi.org/./..

Dynamos at extrememagnetic Prandtl numbers: insights fromshell models

Mahendra K. Verma and Rohit Kumar

Department of Physics, Indian Institute of Technology, Kanpur, India

ARTICLE HISTORYReceived January Accepted August

KEYWORDSMagnetic field generation;dynamo; energy transfers;shell model;magnetohydrodynamicturbulence

ABSTRACTWe present a magnetohydrodynamic (MHD) shell model suitable forcomputation of various energy fluxes of MHD turbulence for verysmall and very large magnetic Prandtl numbers Pm; such computa-tions are inaccessible to direct numerical simulations. For small Pm,we observe that both kinetic and magnetic energy spectra scale ask−5/3 in the inertial range, but the dissipative magnetic energy scalesas k−11/3exp (− k/k

η). Here the kinetic energy at large length scale

feeds the large-scalemagnetic field that cascades to small-scalemag-netic field, which gets dissipated by Joule heating. The large-Pmdynamo has a similar behaviour except that the dissipative kineticenergy scales as k−13/3. For this case, the large-scale velocity field trans-fers energy to the large-scale magnetic field, which gets transferredto small-scale velocity and magnetic fields; the energy of the small-scale magnetic field also gets transferred to the small-scale velocityfield, and the energy thus accumulated is dissipated by the viscousforce.

1. Introduction

One of the celebrated problems in physics and astrophysics is the generation of magneticfield in planets, stars, and galaxies [1]. Themost popular theory for this phenomenon is thedynamo mechanism in which the magnetic field is amplified due to the nonlinear energytransfer from the velocity field to the magnetic field. The magnetic energy first grows andthen saturates if the energy supply to the magnetic field could overcome the Joule dissipa-tion. In this paper, we will report the energy spectra and fluxes during the steady state usinga popular model called shell model.

Some of the important parameters for dynamo are the magnetic Prandtl numberPm= ν/η and themagnetic Reynolds number Rm=UL/η, where ν is the kinematic viscos-ity, η is the magnetic diffusivity, andU, L are the velocity and length scales of the system. Arelated quantity of importance is the kinetic Reynolds number Re = UL/ν = Rm/Pm. Theother important factors for dynamo are related to the rotation rate, temperature, viscousdissipation, geometry, etc.

CONTACT Rohit Kumar rohitkumar.iitk@gmail.com; mkv@iitk.ac.in© Informa UK Limited, trading as Taylor & Francis Group

JOURNAL OF TURBULENCE 1113

The sustained dynamo is possible only for some set of parameters. For example, theenergy inflow to the magnetic field at the large scale should, on the average, compen-sate the losses due to the Joule dissipation. This condition and dimensional analysis yieldUB/L�ηB/L2 or Rm�1. However, numerical simulations and experimental studies indi-cate that the critical magnetic Reynolds number Rmc > 10 [2–4]. Kazantsev theory yieldsa critical Rm of approximately 100. The magnetic Prandtl number also plays a major rolein the dynamo mechanism. Numerical and experimental studies show that the nature ofsmall-Pm and large-Pm dynamos are very different. The planetary and solar dynamos havetypically small Pm (of the order of 10−5), while the interstellar dynamos have large Pm(of the order of 1011) [5,6]. Unfortunately, direct numerical simulation (DNS) of small-Pmor large-Pmdynamos is very difficult due to large-scale separation between the velocity andmagnetic dissipation regimes. At present, the largest and smallest Pm simulated so far areof the order of 100 and 0.01, respectively [7–9]. Schekochihin et al. [7] performed dynamosimulations with large Pm (25 � Pm � 2500), while Haugen et al. [8] and Brandenburg[9] simulated dynamos for 1/8 � Pm � 30 and 0.01 � Pm � 1, respectively. Federrathet al. [10] performed three-dimensional high-resolution magnetohydrodynamic (MHD)simulation of supersonic turbulencewith Pm= 0.1—10 inwhich they observed dynamo forboth high and lowmagnetic Prandtl numbers. They reported that the dynamo growth rate,and the ratio of the magnetic energy and the kinetic energy at the saturation increase withPm. They also compared simulation results with theoretical predictions. Schober et al. [11]presented a theoretical model to study saturation of dynamo for Pm � 1 and Pm � 1;they observed that the saturation level of dynamo depends on the type of turbulence andthe Prandtl number, and that the dynamos for Pm � 1 are more efficient than those forPm � 1.

The MHD shell models are quite handy for simulating dynamos with very large andvery small Pm’s. In a shell model, a single shell represents all the modes of a logarithmi-cally binned shell. Therefore, the number of variables required to simulate large-Pm andsmall-Pm dynamos using a shell model are much smaller than its DNS counterpart. As anexample, we can reach Pm = 10−9 and Pm = 109 using shell models with 76 shells. Thereare a large number of MHD shell models [6,12–19] that have yielded interesting resultsregarding the energy spectra and structure functions. Some models yield Kolmogorov’senergy spectrum E(k) ∼ k−5/3 [13], while others yield Kraichnan–Iroshnikov spectrumE(k) ∼ k−3/2 for the velocity and magnetic fields [15].

In the present paper, we focus on the energy transfers of MHD turbulence using shellmodels. A quantification of energy transfers is critical for understanding the growth andsustenance of the magnetic field, and it has been studied by several researchers [20–22]using DNS, albeit at Prandtl numbers near unity. These energy transfer studies have alsobeen performed by using theoretical tools [23,24]. In this paper, we perform similar analysisfor extreme Prandtl numbers using a new shell model. Earlier, Stepanov and Plunian [17],Plunian et al. [6], and Lessinnes et al. [18] had derived formulas for the energy fluxes in theirMHD shell models. These models capture certain aspects of energy fluxes, but they fail toreproduce all the results of DNS. For example, in Lessinnes et al.’s model [18], most of theenergy fluxes are in good qualitative agreement with the DNS results, but not themagnetic-to-magnetic flux (�B<

B>, defined in Appendix 2). For Pm = 10−3, Lessinnes et al.’s modelyields negative �B<

B>, but DNS by Dar et al. [25], Debliquy et al. [26], Alexakis et al. [27],and Mininni et al. [28] (also see Verma [29]) yield positive �B<

B>. Note that all the above

1114 M. K. VERMA AND R. KUMAR

shell models yield the energy spectrum correctly due to the quadratic structures of thenonlinear terms. Derivation of energy fluxes of MHD turbulence, however, requires morecareful selection of the nonlinear terms, which is the objective of this paper.

In the present paper, we construct a new shellmodel with structures suitable for studyingenergy transfers, as well as keeping all the conservation laws of MHD. We observe that thenature of all the energy fluxes including�B<

B> of thismodel are in good qualitative agreementwith theDNS results [25–29].We study the properties of the energy fluxes during the steadystate for small and large Pm using this new shell model.We observe interesting correlationsbetween the energy fluxes, dissipation rates, and energy spectra. A word of caution is thatour shell model includes only the local nonlinear interactions among velocity andmagneticshells, hence the initial phases of the dynamo growth for large Pm are not captured properlyby our shell model. Inclusion of non-locality in the shell model should be able to addressthis and related issues. Also, we have ignored themagnetic and kinetic helicities in our shellmodel. See Plunian et al. [6] for a detailed discussion on shell models with magnetic andkinetic helicities that modifies the energy transfers significantly [5,6,29].

The structure of the paper is as follows. In Section 2, we describe our MHD shell model.Section 3 contains the simulation details and validation of the shell model. In Sections 4and 5, we present the results of our dynamo simulations with small and large Pm, respec-tively. We summarise our findings in Section 6. Appendices 1 and 2 contain the derivationof energy fluxes for fluid and MHD shell models.

2. Description of our MHD shell model

Our GOY-based shell model for dynamo is

dUn

dt= Nn[U,U ] + Nn[B,B] − νk2nUn + Fn, (1)

dBn

dt= Nn[U,B] + Nn[B,U ] − ηk2nBn, (2)

whereUn andBn represent, respectively, the velocity and themagnetic field variables in shelln, Fn is the velocity forcing applied in shell n, ν is the kinematic viscosity, η is the magneticdiffusivity, and kn = k0λn is the wave number of the nth shell with λ = (

√5 + 1)/2, the

golden mean. The chosen λ satisfies λ2 − λ − 1 = 0. The nonlinear terms Nn[U, U], Nn[B,B], Nn[U, B], and Nn[B, U] correspond to, respectively, −U · �U, B · �B, −U · �B, andB · �U, whereU, B are the velocity andmagnetic fields of theMHD equations [6,18,25,29].We derive the structures of the nonlinear terms of the shell model using the properties ofthe nonlinear transfers among the velocity and magnetic variables.

The term Nn[U, U] causes energy exchange among the velocity modes only, and it con-serves total kinetic energy in this process, hence

�(∑

n

U ∗n Nn[U,U ]

)= 0. (3)

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The B field plays no role in this transfer. The above constraint is the same as that for thefluid shell model (Equation (1) with B = 0) in the inviscid limit. Therefore, we choose thesame Nn[U, U] as the GOY fluid shell model (see e.g. [30]), which is

Nn[U,U ] = −i(a1knU ∗

n+1U∗n+2 + a2kn−1U ∗

n+1U∗n−1 + a3kn−2U ∗

n−1U∗n−2

), (4)

where a1, a2, and a3 are constants. The conservation of the total kinetic energy,Eu = (1/2)�n[|Un|2], in the inviscid limit with B = 0 yields

a1 + a2 + a3 = 0. (5)

In addition, we impose a condition that the kinetic helicity (HK = �n(− 1)n|Un|2kn) isconserved for pure fluid case (B = 0), which yields

a1 − λa2 + λ2a3 = 0. (6)

The nonlinear term Nn[U, B], representing −U · �B of the MHD equation, facilitatesenergy exchange among the magnetic field variables with the velocity field variables actingas helper [18,25,29]. These transfers conserve the total magnetic energy, i.e.

�(∑

n

B∗nNn[U,B]

)= 0. (7)

Keeping in mind the above constraint, we choose the following form for Nn[U, B]:

Nn[U,B] = −i[kn

(d1U ∗

n+1B∗n+2 + d3B∗

n+1U∗n+2

) + kn−1( − d3U ∗

n+1B∗n−1 + d2B∗

n+1U∗n−1

)+ kn−2

( − d1U ∗n−1B

∗n−2 − d2B∗

n−1U∗n−2

)], (8)

where d1, d2, and d3 are constants.The net loss of total energy (kinetic and magnetic) due to the aforementioned energy

exchanges is zero for ν = η = 0 [29]. Hence

�(∑

n

U ∗n Nn[B,B] + B∗

nNn[B,U ]

)= 0, (9)

where the term U ∗n Nn[B,B], which corresponds to [B · �B] · U of the MHD equations,

transfers energy from the magnetic energy to kinetic energy, while B∗nNn[B,U ], which cor-

responds to [B · �U] · B, does just the opposite [18]. To satisfy Equation (9), we choosethe following forms of Nn[B, B] and Nn[B, U]:

Nn[B,B] = −2i(b1knB∗n+1B

∗n+2 + b2kn−1B∗

n+1B∗n−1 + b3kn−2B∗

n−1B∗n−2), (10)

Nn[B,U ] = i[kn

(b2U ∗

n+1B∗n+2 + b3B∗

n+1U∗n+2

) + kn−1(b3U ∗

n+1B∗n−1 + b1B∗

n+1U∗n−1

)+ kn−2

(b2U ∗

n−1B∗n−2 + b1B∗

n−1U∗n−2

)], (11)

1116 M. K. VERMA AND R. KUMAR

where b1, b2, and b3 are constants. Note that the above choices of the nonlinear terms satisfythe conservation of total energy of the MHD shell model

E = Eu + Eb = 12

∑n

[|Un|2 + |Bn|2], (12)

where Eu = (1/2)�n[|Un|2] and Eb = (1/2)�n[|Bn|2] are the total kinetic energy and thetotal magnetic energy, respectively. Note that we need to impose only the constraint ofEquation (5) for the above conservation law. The nonlinear terms Nn[B, B], Nn[U, B], andNn[B,U] conserve the total energy automatically. Also, Equation (5) insures that the kineticenergy is conserved for pure fluid case (B = 0) when ν = 0.

The four nonlinear terms contain nine undetermined constants a1, a2, a3, b1, b2, b3, d1,d2, and d3. We determine their values using constraint equations, two of which are Equa-tions (5) and (6). We also use the other two conservation laws, which are the conservationof the total cross helicity Hc and the total magnetic helicity HM when ν = η = 0 and Fn =0. These quadratic conserved quantities are defined as

Hc = 12�

∑n

UnB∗n (13)

HM =∑n

(−1)n|Bn|2/kn, (14)

The conservation of cross helicity yields

b1 + b2 + b3 = 0, (15)

a1 − b3 − d3 − b2 − d1 = 0 (16)

a2 + d3 − b3 − b1 − d2 = 0 (17)

a3 − b2 + d1 − b1 + d2 = 0, (18)

while the conservation of magnetic helicity yields

λ−2(−d1 − b2) − (−d1 + b2) = 0 (19)

λ−2(−d2 − b1) + λ−1(−d2 + b1) = 0 (20)

λ−1(b3 + d3) − (−d3 + b3) = 0. (21)

Thus we have nine constraints [Equations (5, 6, 15–21], and nine unknowns (a1, a2, a3,b1, b2, b3, d1, d2, d3). However, the determinant of the matrix formed by these equationsis zero. Hence, the solution of the above equations is not unique. One of the solutions that

JOURNAL OF TURBULENCE 1117

satisfies all the above constraints is given as follows:

a1 = λ, a2 = 1 − λ, a3 = −1,

b1 = λ, b2 = 1 + λ

2, b3 = −1 − 3λ

2,

d1 = 5λ2

, d2 = −λ + 2, d3 = −λ

2. (22)

It can be shown that (δa1, δa2, δa3, δb1, δb2, δb3, δd1, δd2, δd3), where δ is a constant,also satisfy the equations. We remark that some free parameters are chosen in almost allthe shell models. Biferale [31] and Ditlevsen [32] attribute the above arbitrariness of theparameter to the freedom for the choice of the time scale. The above factor δ is the arbi-trary prefactor. Note that λ (the golden mean) in the above equations satisfies the equationλ2 − λ − 1 = 0.

The aforementioned nonlinear terms Nn[U, U], Nn[B, B], Nn[U, B], and Nn[B, U]facilitate energy transfers from velocity-to-velocity (U2U), magnetic-to-velocity (B2U),magnetic-to-magnetic (B2B), and velocity-to-magnetic (U2B), respectively, and theyinduce energy fluxes of MHD turbulence that play a critical role in the dynamo mecha-nism. The energy flux �X<

Y>(K) is the rate of transfer of the energy of the field X from theshells inside the sphere of radius K to the field Y outside the sphere. The above flux can bewritten in terms of the energy transfer formulas derived in Appendix 2:

�X<Y>(K) =

∑m≤K

∑n>K

∑p

SYX (n|m|p). (23)

In particular,

�U<U>(K) =

∑m≤K

∑n>K

∑p

SUU (n|m|p) (24)

�B<B>(K) =

∑m≤K

∑n>K

∑p

SBB(n|m|p) (25)

�U<B> (K) =

∑m≤K

∑n>K

∑p

SBU (n|m|p) (26)

�B<U>(K) =

∑m≤K

∑n>K

∑p

SUB(n|m|p). (27)

The energy flux �U<B< (K) is the energy flux from the U-shells inside the sphere to the B-

shells inside the sphere, while �U>B> (K) is the corresponding energy flux for the shells out-

side the sphere. We refer to Appendix 2 for a detailed derivation of energy fluxes.In the following section, we also report the kinetic and magnetic energy spectra which

are defined as

Eu(k) = 12

|Un|2kn

, (28)

1118 M. K. VERMA AND R. KUMAR

Eb(k) = 12

|Bn|2kn

. (29)

In the inertial range, we observe that |Un|2 ∼ |Bn|2 ∼ k−2/3n , hence the energy spectra appear

to scale as k−5/3n , which is the Kolmogorov’s spectrum.

We point out that our shell model differs from earlier ones due to the aforementionedstructures of energy transfers. Biskamp’s shell model [13] uses Elsässer variables. The formof the nonlinear terms of Stepanov and Plunian’s shell model is very different from ours(see Equations (1)–(3) of [17]). In the shell model of Frick et al. [33], dBn/dt does notinvolve Un at all, while the shell model of Plunian and Stepanov [34] includes non-localterms. In the shell model of Stepanov and Plunian [17], Lessinnes et al. [18], and Stepanovand Plunian [35], the structures ofNn[B, B] andNn[U,U] are the same, and so are those ofNn[U, B] andNn[B,U]. As a result, thesemodels do not yield correctmagnetic-to-magneticenergy transfers (B2B) due toNn[U, B], as well as magnetic-to-velocity (B2U) and velocity-to-magnetic (U2B) transfers arising due to Nn[B, B] and Nn[B, U]. This is the reason forthe negative B2B flux arising in these models. As described above, our model correctsthese deficiencies in the previous shell models and it yields correct energy transfers. Wealso remark that earlier shell models reproduce the kinetic and magnetic energy spectraquite correctly since they depend on the quadratic structure of the shell model. The energytransfers, however, require more sophisticated structuring of the nonlinear terms.

In the next section, we provide simulation details.

3. Simulation details and validation for Pm= 1

In our shell model simulations, we divide the wave number space into 36 or 76 logarith-mically binned shells. The large number of shells has been used for simulating dynamoswith Pm= 10−9 and 109. We use the forcing scheme of Stepanov and Plunian [17] in whichthe forcing is applied to three neighbouring shells nf, nf + 1 and nf + 2 as Fn f + j = f jeiφ j

(j = 0, 1, 2). Here fj’s are real positive numbers derived in Stepanov and Plunian [17], andφj � [0, 2π] are random phases. In our simulations, we employ external random forcingto the velocity shells 3, 4, and 5 such that the kinetic energy supply rate is maintained ata constant value (ϵ = 1), and the normalised kinetic and magnetic helicities as well thenormalised cross helicity are relatively small.

We first initiate a pure fluid simulation with a random initial condition for the velocityfield and run the simulation till it reaches a statistically steady state. For the initial con-dition for a dynamo simulation, we take the above steady fluid state as the initial velocityconfiguration and a small seed magnetic field at shells 1 and 2. The dynamo simulation iscarried out till it reaches a steady state. For the time integration, we employ Runge-Kuttafourth order (RK4) scheme with a fixedt. The choice of a fixedt helps during the kine-matic growth phase where dynamic t varies widely. In our forced MHD simulations, thekinetic and magnetic energies saturate at t � 10, but we carry out the simulations for amuch longer time. Here the unit of time is the eddy turnover time. For the computationof the energy spectrum and energy fluxes, we average these quantities (in steady state) forlong time intervals.

JOURNAL OF TURBULENCE 1119

Table . Parameters of shell model simulations (D= decay-ing, F= forced): number of shells (n), kinematic viscosity (ν),magnetic diffusivity (η), magnetic Prandtl number (Pm =ν/η), kinetic Reynolds number (Re = UL/ν), and magneticReynolds number (Rm= UL/η).Sim n ν η Pm Re Rm

D − − .× .×

F − − .× .×

F − − − .× .×

F − − .× .×

F − − − .× .×

F − − .× .×

Table . Parameters of shell model simulations: magnetic Prandtl num-ber, time-averaged values of total kinetic energy (〈Eu〉), total magneticenergy (〈Eb〉), 〈Eu〉/〈Eb〉, total kinetic energy dissipation rate (〈ϵν〉), totalmagnetic energy dissipation rate (〈ϵη〉), 〈ϵν〉/〈ϵη〉, normalised kinetic helicity(〈hK〉 = 〈(�n(− )n|Un|kn)/(�n|Un|kn)〉), normalised magnetic helicity (〈hM〉= 〈(�n(− )n|Bn|/kn)/(�n|Bn|/kn)〉), and normalised cross helicity (〈hc〉 =⟨(� ∑

n 2UnB∗n)/

∑n(|Un|2 + |Bn|2)

⟩).

Sim Pm 〈Eu〉 〈Eb〉〈Eu〉〈Eb〉

〈ϵν〉 〈ϵ

η〉 〈ε

ν〉

〈εη〉 〈hK〉 〈hM〉 〈hc〉

F . . . . . . . . − .F − . . . . . . . . − .F . . . . . . . . − .F − . . . . . . . . .F . . . . → → � . . − .

We perform our simulations for six sets of parameters listed in Table 1. The magneticPrandtl numbers for these runs are Pm = 1, 10−3, 103, 10−9, and 109, thus we cover verysmall to very large Pm’s. Note that the steady-state kinetic Reynolds number and mag-netic Reynolds number are quite large, hence our runs are in the turbulent regime. In ourshell model simulations, the total kinetic and total magnetic energies fluctuate considerablywith time. Therefore, we compute the time-averaged values of the total kinetic energy (Eu),the total magnetic energy (Eb), and their ratio, and present them in Table 2. In the forceddynamo simulations, the fluctuations in Eu/Eb is of the order of unity, whereas in the decay-ing simulation, it is of the order of 10−2. Stepanov and Plunian [17] also observed similarfluctuations in kinetic and magnetic energies in their shell model of dynamo.

We compute the kinetic and magnetic helicities. The normalised kinetic and magnetichelicities, as well as the normalised cross helicity are small (less than a quarter), hence allour runs are non-helical (see Table 2). In Table 2, we also exhibit the kinetic and magneticenergy dissipation rates as well as their ratio. We observe that the ratio of the dissipationrates increases with Pm, from very small values for Pm � 1 to very large values for Pm �1, similar to that observed by Brandenburg [36].

For validation of our flux formulas, we performed decaying and forced shell model sim-ulations for Pm= 1 with 36 shells. In Figure 1, we plot the kinetic energy spectrum |U(k)|2,the magnetic energy spectrum |B(k)|2, and the total energy spectrum |U(k)|2 + |B(k)|2, allof which exhibit an approximate k−2/3 spectrum in correspondence with the Kolmogorov’sspectrum of Eu(k) ∼ Eb(k) ∼ k−5/3 (see Equations (28) and (29)). The aforementioned

1120 M. K. VERMA AND R. KUMAR

Figure . For Pm = : time-averaged kinetic energy (|U(k)|), magnetic energy (|B(k)|), and total energy(|U(k)| + |B(k)|) in spectral space for (a) decaying simulation and (b) forced simulation. See Table fordetails. Figure exhibits the energy fluxes at K corresponding to the vertical solid line. Here kν = (ϵ/ν)/

and kη = (ϵ/η)/.

results have been observed for a large number of shell models [15,17,18,34]. We computethe shell model spectral exponents using linear regression on the log–log plots. The expo-nents with the corresponding errors for various simulations are listed in Table 3. Note thatthe dissipation of Eu(k) starts near Kolmogorov wave number k = kν = (ϵ/ν3)1/4. The cor-responding dissipative wave number for Eb(k) is kη = (ϵ/η3)1/4.

The primary advantage of our shell model is its ability to compute the energy fluxes ofMHD turbulence.We compute the fluxes for the decaying as well as the forced shell modelsby averaging over a long time. For the decaying run, the averaging has beenperformedwhenthe variations in the energy are reasonably small. Various energy fluxes for the forced shellmodel simulation are shown in Figure 2 that shows constant values for these fluxes in theinertial range. In Figure 3(a,b), we exhibit the constant values of the fluxes for the wavenumber sphere of radius K = 123 for the decaying and forced simulations. In Table 4, welist these values along with the DNS results of Dar et al. [25] and Debliquy et al. [26].

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Table . Exponents of the kinetic andmag-netic energy spectra with correspondingerrors.Sim Pm KE exponent ME exponent

D −.± . −.± .F −.± . −.± .F − −.± . −.± .F −.± . −.± .F − −.± . −.± .F −.± . −.± .

Figure . For forced dynamo simulation with Pm = : time-averaged steady-state energy fluxes: �U<U>,

�B<U>, �

U<B< , �

U>B> , �

U<B> , �

B<B>, and �tot = �U<

U> + �U<B> + �B<

U> + �B<B>. Figure exhibits the energy

fluxes at K corresponding to the vertical solid line.

Table . For decaying and forcedMHD simulationwith Pm= :time-averaged energy fluxes, viscous dissipation rate (ϵν ), andJoule dissipation rate (ϵη) forK= in our shellmodel.We alsolist these quantities for the decaying D dynamo simulation byDebliquyet al. [], and for the forcedDdynamosimulationbyDar et al. []. The uncertainty in the values are approximately.. Here− represents unavailable values.

rA = . rA = . rA = . rA = .(Deb, DNS) (Dar, DNS) (shell model) (shell model)(Decaying) (Forced) (Decaying) (Forced)

Flux (K� ) (K= ) (K= ) (K= )

�U<U>

. −. . .�U<

B> . . . .�B<

U>. −. . .

�B<B> . . . .

�U<B< −. . −. .

�U>B> . −. . .

ϵν(U<) − − . .

ϵη(B<) − − . .

ϵν(U>) − . . .

ϵη(B>) − . . .

1122 M. K. VERMA AND R. KUMAR

Figure . For dynamo simulation with Pm = , schematic diagrams of the time-averaged energy fluxesand dissipation rates for the wave number sphere of radius K = : (a) decaying simulation, (b) forcedsimulation. The thick arrow indicates kinetic energy supply rate, whereas thewavy arrows indicate kineticenergy and magnetic energy dissipation rates. The uncertainty in the values are approximately ..

Some of the important features of our flux results are the following:

� The Alfvén ratio rA = Eu/Eb is approximately 0.5 for the decaying simulation towardsits later phase. However, it is approximately 1.5 for the forced run during the steadystate. Note that DNS yields rA � 0.5 for both decaying (at a later stage) and forcedruns.

� For the forced run,�U<B< is the most dominant flux indicating a strong energy transfer

from the large-scale velocity field to the large-scalemagnetic field. The energy receivedby the large-scale (small k) magnetic field is transferred to the small-scale (large k)magnetic field by a forward cascade of magnetic energy (�B<

B> > 0). This result of oursis in good qualitative agreement with that of Dar et al. [25].

� For the forced run, the energy fluxes�U<U>,�U<

B> ,�B<U>,�U>

B> are all positive. The visibledifferences between the DNS and shell model may be because Dar et al. [25] simulatedtwo-dimensional MHD turbulence.

JOURNAL OF TURBULENCE 1123

� For rA � 0.5, �U<B< < 0 implying that the large-scale magnetic field transfers energy

to the large-scale velocity field. This result is in agreement with the DNS results ofDebliquy et al. [26], who argue that in decaying simulations, �U<

B< < 0 for Eb > Eu

and vice versa. Our shell model is consistent with the above observations of Debliquyet al.

� In both decaying and forced shell model simulations, the�B<B> > 0, which is consistent

with the DNS results. The Lessinnes et al.’s shell model [18], however, yields�B<B> < 0.

Hence, our shell model is a better candidate for computing energy fluxes of MHDturbulence than that of Lessinnes et al. We also remark that the Stepanov and Plu-nian’s [17] formula for �B<

B> has certain ambiguities.� The errors in the computed energy fluxes are 2%. Hence the uncertainty in the energybalance, e.g. the difference between the sum of the dissipation rates and the energysupply rate, is approximately 2%.

The above results indicate that our shell model is a good candidate for studying energytransfers in MHD turbulence.

The dynamo mechanism is typically studied for small Pm, mimicking the stellar andplanetary dynamos, and for large Pm corresponding to the astrophysical plasmas. In thefollowing sections, we will describe the energy spectra and fluxes for such dynamos.

4. Dynamowith small Prandtl numbers

We performed dynamo simulations for Pm = 10−3 and 10−9, which are completely inac-cessible to the DNS at present. The numbers of shells used for these runs are 36 and 76,respectively. First, we simulate the fluid shell model (B = 0) till it reaches the steady state.After this, we start our dynamo simulations with the above steady-state velocity field and arandom seedmagnetic field. The dynamo is forced at low wave numbers (shells 3−5) usingthe method of Stepanov and Plunian (described in Section 3).

In Figure 4, we plot the evolution of kinetic andmagnetic energy spectra. For both Pm’s,the kinetic energy does not change appreciably. The magnetic energy spectrum, however,grows first at large wave numbers due to nonlinear transfers, during which an approximatescaling of |Bn|2 ∼ k5/2 or the magnetic energy Eb(k) ∼ k3/2 for the low wave numbers, con-sistent with the spectrum proposed by Kazantsev [37]. Later, |Bn|2 starts to grow at lowerwave numbers. These results are consistent with the earlier DNS results ([5], and referencestherein).

In Figure 5(a,b), we plot the steady-state energy spectra for Pm = 10−3 and 10−9. ForPm = 10−3, we observe that Eu(k) ∼ Eb(k) ∼ k−5/3 curve fits reasonably well in the inertialrange, consistent with earlier simulations [17,18,34]. For both Pm’s, Eu(k)∼ k−5/3 for k�kν ,but Eb(k) ∼ k−5/3 for k�kη, and Eb(k) ∼ k−11/3exp (− k/kη) for kη�k�kν , as deduced byBatchelor et al. [38] and Odier et al. [39]; the dissipative spectra will be described later inthis section.

We compute various energy fluxes for both the Prandtl numbers. In the early stages, weobserve a rapid energy transfer to the large-k magnetic fields, but in the steady state weobserve two broad regimes of energy fluxes as shown in Figure 6(a,b). The first regime,k < kη, corresponding to the Kolmogorov’s k−5/3 regime for both U and B fields, exhibits

1124 M. K. VERMA AND R. KUMAR

Figure. FordynamosimulationwithPm= −: evolutionof kinetic energy (dashed lines) andmagneticenergy (solid lines) spectra with time. The seed magnetic field at t = was at the shells and . Themagnetic field grows first at large wave numbers and later at small wave numbers. In the early phase(kinematic regime), |Bn|

∼ k/ or Eb(k)∼ k/ which corresponds to the Kazantsev scaling.

Table . Time-averaged energy fluxes, viscous dissipation rate,and Joule dissipation rate for Pm= − for our shell model. Wealso list the results of the shell model by Lessinnes et al. [] forPm= −.

Pm= −

Lessinnes shell model Our shell model Our shell modelFlux (K� ) (K= ) (K= .× )

�U<U>

. . .�U<

B> . . .�B<

U>. . .

�B<B> −. . .

�U<B< . . .

�U>B> . . .

ϵν(U<) . . .

ϵη(B<) . . .

ϵν(U>) . . .

ϵη(B>) . . .

similar behaviour as Pm = 1 case. The energy supplied at the small-k of U field getstransferred to the small-kmagnetic field, B<, and the large-kmagnetic field, B>. In addi-tion, we have a forward cascade of themagnetic energy that leads to a significant build-up ofmagnetic energy near the wave numbers k � kη (including a bump). The magnetic energyis dissipated by Joule heating in the second regime, kη < k < kν .

In Table 5, we show time-averaged energy fluxes for Pm = 10−3 at K = 123 and 4.39 ×105. For comparison, we also include the shell model results of Lessinnes et al. [18] forPm = 10−3. We observe that the energy flux �U<

U> in our simulation is significantly smallerthan that of the Lessinnes et al. The other major difference is that the energy flux �B<

B>

in our simulation is positive, whereas it is negative for Lessinnes et al, i.e. they observedinverse cascade of the magnetic energy. In our shell model simulations, we do not observeany inverse cascade of the magnetic energy, which is consistent with the DNS results ofKumar et al. [22]. As described earlier, the difference in the �B<

B> flux arises due to different

JOURNAL OF TURBULENCE 1125

Figure . For dynamo simulation with (a) Pm = − and (b) Pm = −: time-averaged kinetic energy,magnetic energy, and total energy spectra. The dashed vertical lines represent Kolmogorov wave num-bers kν and kη . For Pm= −, Figure exhibits the energy fluxes at K’s corresponding to the vertical solidlines.

form of nonlinearity in the twomodels. In Table 6, we show energy fluxes for Pm= 10−9 atK = 123 and 1.15 × 106.

As shown in Figure 7(b) and Table 6, for k�kη, �U<B< ≈ 97% of the total energy feed;

this magnetic energy gets dissipated into Joule heating. A small fraction of input energyis transferred to small-scale velocity field via �U<

U> (�6%) that gets dissipated via viscousdamping. In summary, most of the input energy is transferred to the magnetic field thatgets dissipated by Joule heating.

Now we describe the phenomenology of small-Pm MHD turbulence. In the inertialrange where nonlinear terms of both Equations (1) and (2) dominate, the energy spectrumfollows Kolmogorov-like spectrum [40,41]. However, for kη�k�kν , the kinetic energy con-tinues to have k−5/3 energy spectrum, but the magnetic energy spectrum becomes steeper.Odier et al. [39] deduced that Eb(k) ∼ k−11/3 using scaling arguments similar to that of

1126 M. K. VERMA AND R. KUMAR

Figure . For dynamo simulationwith (a) Pm= − and (b) Pm= −: time-averaged energy fluxes. (b)has the same legends as (a). The inset contains the plot of�all

B> = �B<B> + �U<

B> + �U>B> . For Pm= −,

Figure exhibits the energy fluxes at K’s corresponding to the vertical solid lines.

Table . Time-averaged energy fluxes, vis-cous dissipation rate, and Joule dissipationrate for Pm= − for our shell model.

Pm= −

Our shell model Our shell modelFlux (K= ) (K= .× )

�U<U>

. .�U<

B> . .�B<

U>. .

�B<B> . .

�U<B< . .

�U>B> . .

ϵν(U<) . .

ϵη(B<) . .

ϵν(U>) . .

ϵη(B>) . .

JOURNAL OF TURBULENCE 1127

Figure . For dynamosimulationwithPm= −: schematic diagramsof the time-averagedenergyfluxesand dissipation rates for K= and K= .× . The wavy arrows represent viscous and Joule dissipa-tion rates.

Batchelor et al. [38]. In the regime kη�k�kν , Odier et al. [39] matched Nn(B, U) and theJoule dissipation term of Equation (2) that yields

BkηkUk ∼ ηk2Bk. (30)

In the language of energy transfers, �B<B> is of the same order as the local Joule dissipation

ηk2|Bk|2 [25]:

kBkηUkBk ∼ ηk2B2k. (31)

The kinetic energy continues to follow K−5/3 spectrum due to the strong Nn(U, U) term.Using Equation (30) andU 2

k /k ∼ ε2/3k−5/3, Odier et al. [39] deduced that

Eb(k) = B2k

k∼

(Bkη

ηk

)2 U 2k

k

∼(Bkη

η

)2

ε2/3k−11/3. (32)

1128 M. K. VERMA AND R. KUMAR

Our numerical data, however, show steeper spectrum than k−11/3 due to the Joule dissipa-tion ηk2Eb(k). To account for the dissipation, we modify the above spectrum to

Eb(k) ∼(Bkη

η

)2

ε2/3k−11/3 exp(−k/kη), (33)

which matches with the numerical data quite well, as shown in Figure 5. Monchauxet al. [42] report k−11/3 and k−17/3 spectra at different regimes of Eb(k) of the von Kármánsodium experiment. They attribute the difference to the advection of the magnetic fieldeither by eddies of the length scales 1/kη or 1/k.

Since the magnetic energy flux is dissipated by the Joule dissipation, we obtain [43]

d�allB>

dk= −2ηk2Eb(k), (34)

where �allB> = �B<

B> + �U<B> + �U>

B> is the total energy flux to the magnetic field at smallscales. Using Equation (33), we can argue that

�allB>(k) ∼ 2ηk3Eb(k) ∼ k−2/3 exp(−k/kη). (35)

We observe that �allB>(k) is in qualitative agreement with the above scaling (see inset of

Figure 6(b)).In the next section, wewill simulate dynamoswith large Prandtl numbers using our shell

model.

5. Dynamowith large Prandtl numbers

Schekochihin et al. [7], Stepanov and Plunian [17], Buchlin [44], Kumar et al. [21], andothers showed that the non-local energy transfers play amajor role in such dynamos duringthe dynamo growth phase. Critical non-local interactions are missing in our shell model.Still, we use the local shell model described in Section 2 to simulate large-Pm dynamo andexplore how well the local shell model captures the dynamo growth and magnetic energyfluxes of large-Pm dynamo.

We performed dynamo simulations for Pm= 103 and 109. Similar to small Pm runs, westart our dynamo simulation with the steady-state velocity field of the fluid shell model anda random seedmagnetic field. Themagnetic energy first spreads to the larger wave number,and then it grows at smaller wave number as well. The evolution of the energy spectrum,shown in Figure 8, has strong similarities with the small-Pm dynamo. Initially, the mag-netic energy is quickly transferred from small wave numbers to larger wave numbers vialocal transfers. After sometime, themagnetic energy grows at large k (small scales) possiblydue to weaker magnetic diffusion. Thus, some features of the kinematic growth and satu-ration are captured by the local shell model. However, non-local energy transfers from thesmall-wave number velocity shells to large-wave number magnetic shells are missing in theevolution, which are captured only by non-local shell models [35].

We studied the energy spectra for the steady-state dynamos with Pm = 103 and 109.We illustrate these spectra in Figure 9(a,b). Note that kν < kη for these runs. The plots of

JOURNAL OF TURBULENCE 1129

Figure . For dynamo simulationwith Pm= : evolution of kinetic energy (dashed lines) andmagneticenergy (solid lines) spectra with time. |Bn|

∼ k/ corresponds to the Kazantsev prediction at the earlystages.

Figure 9 reveal that Eu(k) ∼ Eb(k) ∼ k−5/3 in the inertial range with k�kν . However, forkν�k�kη, the magnetic energy spectrum continues to scale as Eb(k) ∼ k−5/3, but Eu(k) ∼k−13/3 due to the following reasons.

For large Pm, the momentum equation can be approximated by

�������∂U∂t

+ U · ∇U = −∇p+ B · ∇B + ν∇2U, (36)

because the viscous term dominates the flow. Here p is the pressure. Performing dimen-sional analysis, we obtain

kB2k ∼ νk2Uk. (37)

Using Eb(k) = B2k/k ∼ ε2/3k−5/3 and the above expression, the kinetic energy spectrum is

Eu(k) ∼ U 2k

k∼ ε4/3

ν2 k−13/3. (38)

Hence we expect from the shell model that |Uk|2 ∼ k−10/3, which is nicely borne out inour simulation results with |Uk|2 ∼ k−3.3 ± 0.2. The above scaling is very similar to those ofPandey et al. [45] derived for turbulent convection under infinite thermal Prandtl number.

We compute various energy fluxes for both the runs during the steady state by averagingthem over a long time. The simulation results are exhibited in Figure 10. The energy fluxesfor both the Prandtl numbers appear to be similar. The fluxes take constant values in theinertial range, after which all the fluxes tend to zero. The net energy flux to the velocityfield, �all

U> = �U<U> + �B<

U> + �B>U>, is dissipated by the viscous dissipation, hence

d�allU>

dk= −2νk2Eu(k). (39)

1130 M. K. VERMA AND R. KUMAR

Figure . For dynamo simulation with (a) Pm = and (b) Pm = : time-averaged kinetic energy,magnetic energy, and total energy spectra. For Pm= , Figure exhibits the energy fluxes at K’s corre-sponding to the vertical solid lines.

Using Equation (38), we can argue that

�allU> ∼ 2νk3Eu(k) ∼ k−4/3, (40)

which is qualitatively borne out by �allU> as well dominant fluxes as shown in Figure 10.

In Figure 11, we display the constant values of the energy fluxes for Pm= 109 atK= 123andK= 1.15× 106. The kinetic energy at small wave number is transferred to B< andB>,which in turn gets transferred to U > (large wave number). This energy is dissipated viaviscous damping. Interestingly, the direct cascade of kinetic energy to small scales (�U<

U>)is very small; the growth of the kinetic energy at small scales occur via kinetic-to-magneticand then back to kinetic energy transfers. Almost all the energy fed at small wave numbersof the velocity field is dissipated at large wave numbers of the velocity field by viscous dis-sipation. The Joule dissipation is insignificant for large-Pm dynamo. In contrast, maximaldissipation in small-Pm dynamos occurs via Joule heating.

JOURNAL OF TURBULENCE 1131

Figure . For dynamo simulation with (a) Pm = and (b) Pm = : time-averaged energy fluxes. (b)has the same legends as (a). The inset contains the plot of �all

U> = �U<U> + �B<

U> + �B>U>. For Pm = ,

Figure exhibits the energy fluxes at K’s corresponding to the vertical solid lines.

In Table 7, we show time-averaged energy fluxes for Pm = 103 at K = 123, and forPm = 109 at K = 123 and 1.15 × 106. For comparison, we also list DNS results for Pm =20 by Kumar et al. [21]. The energy fluxes in our shell model simulation are in qualita-tive agreement with the DNS results of Kumar et al., but with certain major differences. Inour shell model, �U<

B< is the most dominant energy flux for the magnetic energy growth,whereas in DNS the magnetic energy grows primarily due to �U<

B> . The deviation fromKumar et al. is due to the absence of non-local interactions in our shell model. A non-localshell model should be able to capture the aforementioned non-local energy transfers.

We summarise our spectra results by plotting Eb(k)/Eu(k) for various Prandtl numbers(see Figure 12). We observe that the Eb(k)/Eu(k) � 1.5 in the regime where both Eu(k) ∼Eb(k) ∼ k−5/3. For large Pm, Eb(k)/Eu(k) grows monotonically with k, but it decreases withk for small Pm due to the nature of energy spectra described in the last two sections. Ourresults are in general agreement with the results of Stepanov and Plunian [35]. Note that theinertial range in our simulation is broader because our viscosity and themagnetic diffusivityare smaller than those of Stepanov and Plunian [35].

1132 M. K. VERMA AND R. KUMAR

Figure . For dynamo simulation with Pm= : schematic diagram of the time-averaged energy fluxesand dissipation rates at K= and K= .× .

Table . Time-averaged energy fluxes, viscous dissipation rate, andJoule dissipation rate for Pm= and for our shell model. Wealso list the DNS results of Kumar et al. [] for Pm= .

Pm= Pm= Pm=

DNS (Kumar) Shell model Shell model Shell modelFlux (K� ) (K= ) (K= ) (K= .× )

�U<U>

. . . .�U<

B> . . . .�B<

U>. . . .

�B<B> . . . .

�U<B< . . . .

�U>B> −. −. −. .

ϵν(U<) − . . .

ϵη(B<) − . . .

ϵν(U>) − . . .

ϵη(B>) − . . .

JOURNAL OF TURBULENCE 1133

Figure . For Pm� [−, ]: the ratio ofmagnetic energy (|B(k)|) and kinetic energy (|U(k)|)withwavenumber. We also plot the ratio computed for Pm= − and by Stepanov and Plunian (S & P) [].

6. Discussion and conclusions

In this paper, we presented an MHD shell model that is suitable for computing energyfluxes. We derived formulas for various fluxes of dynamo. The fluxes of our shell modelare in good agreement with those computed using DNSs for the magnetic Prandtl num-ber, Pm = 1 [26]; thus we validate the shell model for flux computations. We employ thismodel to study energy spectra and fluxes for very small and very large magnetic Prandtlnumbers (from Pm= 10−9 to 109), which are inaccessible to DNSs. Our major findings areas follows:

(1) For small Pm, the kinetic energy spectrum Eu(k) ∼ k−5/3, but the magnetic energyspectrum Eb(k) ∼ k−5/3 for k < kη = (ϵ/η3)1/4, and Eb(k) ∼ k−11/3exp (− k/kη) forkη < k < kν , where kν = (ϵ/ν3)1/4, similar to Odier et al. [39].

(2) For small Pm, small-k velocity field feeds small-k magnetic field that cascades tolarge-k magnetic field where it gets dissipated. Here, most of the dissipation isthrough Joule heating.

(3) The large Pmdynamo exhibitsEb(k)∼ k−5/3, butEu(k)∼ k−5/3 for k< kν andEu(k)∼k−13/3 for kν < k < kη.

(4) For large Pm, small-k velocity field feeds the small-k magnetic field, which in turncascades to large-k velocity and magnetic field. The large-kmagnetic field transfersenergy to the large-k velocity field. As a result, most of the energy fed to the velocityfield by forcing at small-k returns to large-k velocity field where it gets dissipated viaviscous force.

In conclusion, we construct an MHD shell model which we use to compute the energyspectra and fluxes for extreme Pm’s. The numerical simulation of this model provides valu-able insights into small-Pm and large-Pm dynamos. Extensions of our shell model withnon-local and realistic helical effects would be very useful for more realistic dynamos.

1134 M. K. VERMA AND R. KUMAR

Acknowledgements

We thank Abhishek Kumar, Daniele Carati, Thomas Lessinnes, Rodion Stepanov, Franck Plunian,Stephan Fauve, and Sagar Chakraborty for discussions at various stages, and A. G. Chatterjee for thehelp with post-processing scripts. This work was supported by a research grant SERB/F/3279/2013-14 from Science and Engineering Research Board, India.

Disclosure statement

No potential conflict of interest was reported by the authors.

Funding

This work was supported by a research grant [grant number SERB/F/3279/2013-14] from Scienceand Engineering Research Board, India.

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1136 M. K. VERMA AND R. KUMAR

Appendix 1. Energy transfer and fluxes in fluid shell model

Our objective is to derive an expression for the energy fluxes inMHD turbulence for whichwe need to derive a formula for the shell-to-shell energy transfer. For simplicity, first wederive these formulas for the fluid shell model, which is Equation (1) with B = 0.

As discussed in Section 2, the kinetic energy of this system is conserved when ν = 0 andFn = 0. However, we get further insights into the energy transfers when we focus on a singleunit of energy transfers, which is a set of three consecutive shells (n − 1, n, n + 1). Notethat this unit is analogous to a triad interaction (k, p, q) with k = p + q in Fourier spacerepresentation of theNavier–Stokes equation. The energy equations for the aforementionedthree shells are

d|Un−1|2/2dt

= −a1kn−1�(Un−1UnUn+1)

= LUU (n − 1|n, n + 1), (A1)

d|Un|2/2dt

= −a2kn−1�(Un−1UnUn+1)

= LUU (n|n − 1, n + 1), (A2)

d|Un+1|2/2dt

= −a3kn−1�(Un−1UnUn+1)

= LUU (n + 1|n − 1, n), (A3)

where � denotes the imaginary part of the argument. Interestingly, the total kinetic energyis conserved in this unit interaction when we impose a1 + a2 + a3 = 0 (Equation (5)). Thisis the detailed energy conservation in a triadic unit of shells.When all the triads are included,then the total energy would also be conserved.

Each of the shells (n− 1, n, n+ 1) receives energy from the other two. Motivated by themode-to-mode energy transfer formulas derived by Dar et al. [25] and Verma [29], we setout to explore whether we can derive a formula for the energy transfer to a shell from theother shells. Let us postulate that SUU(n|m|p) is the energy transfer rate from the shellm tothe shell n with the shell p acting as a mediator. Hence,

SUU (n + 1|n|n − 1) + SUU (n + 1|n − 1|n) = LUU (n + 1|n − 1, n), (A4)

SUU (n|n + 1|n − 1) + SUU (n|n − 1|n + 1) = LUU (n|n − 1, n + 1), (A5)

SUU (n − 1|n + 1|n) + SUU (n − 1|n|n + 1) = LUU (n − 1|n, n + 1). (A6)

Also, the energy received by shell n from the shell m is equal and opposite to the energyreceived by shellm from the shell n. This condition provides three additional relations forthe SUU’s, i.e.

SUU (n + 1|n|n − 1) = −SUU (n|n + 1|n − 1) (A7)

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Figure A. Schematic diagram for energy transfers in a triad.

SUU (n|n − 1|n + 1) = −SUU (n − 1|n|n + 1) (A8)

SUU (n − 1|n + 1|n) = −SUU (n + 1|n − 1|n). (A9)

The desired energy transfer formulae S’s are linear functions ofUn − 1,Un, andUn + 1 as wellas one of the wave number, say kn − 1. Hence, we choose the following form for S’s:

SUU (n + 1|n|n − 1) = α1An, (A10)

SUU (n|n − 1|n + 1) = α2An, (A11)

SUU (n − 1|n + 1|n) = α3An, (A12)

where

An = kn−1�(Un−1UnUn+1). (A13)

These transfers are depicted in Figure A1. Using Equations (A4–A9) and the definitions ofS’s, we obtain

α1 − α3 = −a3, (A14)

α2 − α1 = −a2 (A15)

Hence the solutions of SUU’s are

SUU (n + 1|n|n − 1) = α1An, (A16)

SUU (n|n − 1|n + 1) = (α1 − a2)An, (A17)

SUU (n − 1|n + 1|n) = (α1 + a3)An. (A18)

For our computations, we take α1 = λ = (√5 + 1)/2 after which α2 and α3 are auto-

matically determined.

1138 M. K. VERMA AND R. KUMAR

The above solution, however, is not unique; one can add a circulating transfer that tra-verses (n − 1) → n → (n + 1) → (n − 1). Following the same arguments as Dar et al. [25]andVerma [29], we can show that the circulating transfer does not affect the value of energyflux, which is defined below. A shell model contains a large number of shells. However, allthe energy transfers can be split into the unit interactions discussed above.

The energy flux in fluid turbulence is defined as the energy leaving awave number sphereof radius K. It is easy to define the energy flux �U<

U>(K) for a shell model as the energytransfers from all the shells within a sphere of radius K to the shells outside the sphere,which is

�U<U>(K) =

∑m≤K

∑n>K

∑p

SUU (n|m|p). (A19)

Note that the giver shell is the superscript of � and the receiver shell is the subscript.Since the interactions in such a shell model is local, that is, only three neighbouring shellsinteract, the above energy flux gets contributions from shells K − 1, K, K + 1, K + 2 asgiven below:

�U<U>(K) =

∑n>K

∑m≤K

∑p

SUU (n|m|p)

= SUU (K + 1|K|K − 1) + SUU (K + 1|K|K + 2)+SUU (K + 1|K − 1|K) + SUU (K + 2|K|K + 1)

= α1kK−1�(UK+1UKUK−1) + α2kK−1�(UKUK−1UK+1)

−α3kK−1�(UK+1UK−1UK ) − α3kK−1�(UK+1UK−1UK ) (A20)

In the next appendix, we will extend this formalism to MHD turbulence.

Appendix 2. Energy transfer and fluxes in MHD shell model

The derivation for the energy transfers in MHD turbulence is very similar to that forfluid turbulence described in Appendix 1. It gets more complex due to further interactionsbetween the velocity and magnetic fields. In this appendix, we will state the final resultsafter a sketch of the proof. The derivation follows a similar structure as followed by Daret al. [25] and Verma [29]. In Figure B1, we exhibit a schematic diagram for triadic inter-actions among velocity and magnetic modes (taken from Plunian et al. [6]).

To derive the shell-to-shell energy transfers in MHD turbulence, we limit ourselves to aunit of energy transfers, which is a combination of three consecutive shells (n− 1, n, n+ 1).We start with Equations (1) and (2) and derive the energy equations for the shell variables.The energy equations for the nth shell are

d|Un|2/2dt

= −a2kn−1�(Un−1UnUn+1) − 2b2kn−1�(UnBn−1Bn+1)

= LUU (n|n − 1, n + 1) + LUB(n|n − 1, n + 1), (B1)

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FigureB. Triadic interactions among velocity andmagnetic shells. Reprinted fromPlunian et al. [] copy-right (), with permission from Elsevier.

d|Bn|2/2dt

= b3kn−1�(Bn−1BnUn+1) + b1kn−1�(Un−1BnBn+1)

+d3kn−1�(Bn−1BnUn+1) − d2kn−1�(Un−1BnBn+1)

= LBU (n|n − 1, n + 1) + LBB(n|n − 1, n + 1). (B2)

Here LUU, LBB, LUB, and LBU represent the energy transfers from velocity-to-velocity (U2U),magnetic-to-magnetic (B2B), magnetic-to-velocity (B2U), and velocity-to-magnetic(U2B), respectively.

The LUU terms follow the same dynamics as that for the fluid turbulence, and hence theformulas derived in Appendix 1 is applicable for the U2U transfers. It is easy to show thatthe B2B transfer terms arising due to Nn[U, B] has similar properties as Nn[U, U], that is,it conserves �n|Bn|2/2. Following similar lines as the derivation for fluid shell model, wederive the shell-to-shell transfers for B2B transfers as

SBB(n + 1|n|n − 1) = d2PBBU (n + 1|n|n − 1), (B3)

SBB(n|n − 1|n + 1) = d3PBBU (n|n − 1|n + 1), (B4)

SBB(n − 1|n + 1|n) = (−d1)PBBU (n − 1|n + 1|n), (B5)

where

PYXZ (n|m|p) = kmin(n,m,p)�(YnXmZp), (B6)

with the giver shell X= B, the receiver shell Y= B, and the mediator shell Z=U. Note thatthe velocity field acts as a mediator in B2B energy transfers, and it has similar role as thevelocity field U in the nonlinear interactions of U · �B in the MHD equations.

1140 M. K. VERMA AND R. KUMAR

Using theB2B shell-to-shell formulawe can derive the energy flux�B<B>(K), themagnetic

energy transfers from all the shells within the sphere of radius K to the shells outside thesphere:

�B<B>(K) =

∑m≤K

∑n>K

∑p

SBB(n|m|p). (B7)

Nowwework on theU2B and B2U energy transfers that occur via nonlinear termsNn[B,U] andNn[B, B] of the shell model. The corresponding terms in the energy Equations (B1)and (B2) are LUB and LBU, respectively. It is easy to verify that

LBU (n|n − 1, n + 1) + LUB(n|n − 1, n + 1)+ LBU (n − 1|n, n + 1) + LUB(n − 1|n, n + 1)+ LBU (n + 1|n − 1, n) + LUB(n + 1|n − 1, n) = 0. (B8)

Thus �n(|Un|2 + |Bn|2)/2 is conserved due to these energy transfers; the energy isexchanged among the velocity and magnetic fields. Again following the same procedureas outlined in Appendix 1, we derive the shell-to-shell energy transfers formulas for U2Btransfers as

SBU (n + 1|n|n − 1) = b2PBUB (n + 1|n|n − 1), (B9)

SBU (n + 1|n − 1|n) = b1PBUB (n + 1|n − 1|n), (B10)

SBU (n|n + 1|n − 1) = b3PBUB (n|n + 1|n − 1), (B11)

SBU (n|n − 1|n + 1) = b1PBUB (n|n − 1|n + 1), (B12)

SBU (n − 1|n|n + 1) = b2PBUB (n − 1|n|n + 1), (B13)

SBU (n − 1|n + 1|n) = b3PBUB (n − 1|n + 1|n), (B14)

where PBUB is defined in Equation (B6). Here the giver shell X = U, the receiver shell Y =

B, and the mediator shell Z = B. Also, by definition the energy gained by B from U is theenergy lost by U to B, hence

SUB(n + 1|n|n − 1) = −SBU (n|n + 1|n − 1). (B15)

Other SUB’s can be written in the same manner.Using the shell-to-shell energy transfers, we can write down the energy flux,�U<

B> , whichis the net energy transfer from the shells inside the U sphere of radius K to the shells ofoutside the B sphere, which is

�U<B> (K) =

∑m≤K

∑n>K

∑p

SBU (n|m|p). (B16)

JOURNAL OF TURBULENCE 1141

Similarly, the energy flux, �B<U>, which is the net energy transfer from the shells inside the

B sphere of radius K to the shells of outside the U sphere, which is

�B<U>(K) =

∑m≤K

∑n>K

∑p

SUB(n|m|p). (B17)

We also have energy transfers from the U-shells inside the sphere to the B-shells inside thesphere, which is

�U<B< (K) =

∑m≤K

∑n≤K

∑p

SUB(n|m|p). (B18)

Similarly, for the U2B energy transfer formula for the shells outside the sphere is

�U>B> (K) =

∑m>K

∑n>K

∑p

SUB(n|m|p). (B19)